Mix milk, half-and-half, sugar, cocoa powder, bittersweet chocolate, coffee-flavored liqueur, and espresso powder together in a heavy saucepan; bring to a simmer, stirring often, until heated through, about 5 minutes. Remove saucepan from heat and stir in bananas. Refrigerate until completely cooled, at least 30 minutes.
Whisk yogurt into banana mixture; transfer to the bowl of an ice cream maker. Follow manufacturer's instructions for making frozen yogurt.